Gapless Edge States Research Articles

Topological phase transitions and flat bands on a square-octagon lattice

We construct a square-octagon lattice model by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-nearest-neighbor (NNN) hoppings, where zero-Chern-number topological insulator (ZCNTI) phases emerge. At 1/4 filling, by tuning the staggered fluxes and NNN hopping potential, this model supports the phase transition from a Chern insulator (CI) with Chern number C = 2 to a ZCNTI phase. At half filling, we observe that staggered magnetic fluxes can induce a higher-order topological insulator (HOTI) state. Interestingly, the ZCNTI appears at 3/4 filling when considering the NN and NNN hoppings in the absence of staggered fluxes, which has been rarely reported in previous work. Contrary to conventional HOTIs, this ZCNTI phase hosts both robust corner states and gapless edge states which can be identified based on the quantized dipole and quadrupole moments. Additionally, a topological flat band (TFB) with flatness ratio about 24 appears. We further investigate ν = 1/2 fractional Chern insulator (FCI) state when hard-core bosons fill into this TFB model.

Robust topological insulating property in C2X-functionalized III-V monolayers

Two-dimensional topological insulators (TIs) show great potential applications in low-power quantum computing and spintronics due to the spin-polarized gapless edge states. However, the small bandgap limits their room-temperature applications. Based on first-principles calculations, a series of C2X (X = H, F, Cl, Br and I) functionalized III-V monolayers are investigated. The nontrivial bandgaps of GaBi-(C2X)2, InBi-(C2X)2, TlBi-(C2X)2and TlSb-(C2X)2are found to between 0.223 and 0.807 eV. For GaBi-(C2X)2and InBi-(C2X)2, the topological insulating properties originate from thes-px,yband inversion induced by the spin-orbital coupling (SOC) effect. While for TlBi-(C2X)2and TlSb-(C2X)2, the topological insulating properties are attributed to the SOC effect-induced band splitting. The robust topological characteristics are further confirmed by topological invariantsZ2and the test under biaxial strain. Finally, two ideal substrates are predicted to promote the applications of these TIs. These findings indicate that GaBi-(C2X)2, InBi-(C2X)2, TlBi-(C2X)2and TlSb-(C2X)2monolayers are good candidates for the fabrication of spintronic devices.

Lieb-Schultz-Mattis theorems for symmetry-protected topological phases

The Lieb-Schultz-Mattis (LSM) theorem and its generalizations are a class of powerful no-go theorems that rule out any short-range-entangled (SRE) symmetric ground state irrespective of the specific Hamiltonian, based only on certain microscopic inputs, such as symmetries and particle filling numbers. In this work, we introduce and provide physical arguments for a new class of LSM-type theorems, where any symmetry-allowed SRE ground state must be a symmetry-protected topological (SPT) phase with robust gapless edge states, such as topological insulators and superconductors. The key ingredient is to replace the lattice translation symmetry in usual LSM theorems by the magnetic translation symmetry. These theorems provide new insights into realistic models and experimental realizations of SPT phases in interacting bosons and fermions.

Dimensionality-mediated type-II Dirac semimetal to quantum spin Hall insulator phase transition in TiC

We present first-principles results on monolayer (ML) and bulk titanium carbide (TiC) exhibiting dual topological characteristics: a quantum spin Hall insulator state in two-dimensional ML and a type-II Dirac semimetal state in its three-dimensional bulk form. The nontrivial nature of ML TiC is confirmed through the calculation of a Z2 invariant, spin Hall conductivity, and edge states. The edge band structure of ML TiC displays a single pair of gapless edge states at the M point. The insulating topological phase in ML TiC is driven by a band inversion around the M point involving Ti-d orbitals, with a nontrivial band gap of 0.47 eV. Our findings also indicate that ML TiC possesses excellent dynamical and thermal stability. Moreover, the bulk, ML hollow sphere arrays, and a 4-nm-thick film of TiC have already been synthesized, suggesting the high feasibility of the experimental synthesizing of ML TiC. On the other hand, we demonstrate that the bulk TiC hosts both a type-II Dirac cone and a topological nodal surface without spin-orbit coupling, protected by a combined symmetry of inversion and time reversal. The presence of spin-orbit coupling removes the nodal surface while preserving the type-II Dirac cone. Surface states connecting bulk Dirac nodes in bulk TiC can be readily observed, making the surface characteristics easily detectable in experiments. Published by the American Physical Society 2024

Large-Gap Quantum Spin Hall Insulators in Two-Dimensional Hafnium Halides: Unraveling the Impact of Strain and Substrate.

Two-dimensional (2D) topological insulators (TIs) or quantum spin Hall (QSH) insulators, characterized by insulating 2D electronic band structures and metallic helical edge states protected by time-reversal symmetry, offer a platform for realizing the quantum spin Hall effect, making them promising candidates for future spintronic devices and quantum computing. However, observing a high-temperature quantum spin Hall effect requires large-gap 2D TIs, and only a few 2D systems have been experimentally confirmed to possess this property. In this study, we employ first-principles calculations, combined with a structural search based on an evolutionary algorithm, to predict a class of 2D QSH insulators in hafnium halides, namely, HfF, HfCl, and HfBr with sizable band gaps of 0.12, 0.19, and 0.38 eV. Their topological nontrivial nature is confirmed by a Z2 invariant which equals to 1 and the presence of gapless edge states. Furthermore, the QSH effect in these materials remains robust under biaxial tensile strain of up to 10%, and the use of h-BN as a substrate effectively preserves the QSH states in these materials. Our findings pave the way for future theoretical and experimental investigations of 2D hafnium halides and their potential for realizing the QSH effect.

First principles prediction of novel quantum topological insulator state in two-dimensional XMg2Bi2 (X=Eu/Yb)

Topological insulator (TIs), a novel quantum state of materials, has a lot of significance in the development of low-power electronic equipments as the conducting edge states display exceptional endurance against back-scattering. The absence of suitable materials with high fabrication feasibility and significant nontrivial bandgap, is now the biggest hurdle in their potential applications in devices. Here, we illustrate using first principles density functional calculations that the quintuplet layers of EuMg2Bi2 and YbMg2Bi2 crystals are potential two-dimensional TIs with a sizeable nontrivial gaps of 72 meV and 147 meV respectively. Dynamical stability of these quintuplet layers of EuMg2Bi2 and YbMg2Bi2 is confirmed by our phonon calculations. The weakly coupled layered structure of parent compounds makes it possible for simple exfoliation from a three-dimensional structure. We observed gapless edge states inside the bulk band gap in both the systems which indicate their TI nature. Further, we observed the anomalous and spin Hall conductivities to be quantized in two dimensional EuMg2Bi2 and YbMg2Bi2 respectively. Our findings predict two viable candidate materials as two dimensional quantum TIs which can be explored by future experimental investigations and possible applications of quantized spin and anomalous Hall conductance in spintronics.

Quantum spin Hall insulating phase in two-dimensional MA2Z4 materials: SrTl2Te4 and BaTl2Te4

With the recent synthesis of two-dimensional (2D) MoSi2N4, the 2D material family with the general formula MA2Z4 has become increasingly popular. However, their topological properties have yet to be explored. Using first-principles calculations, we examine the electronic and topological properties of monolayer MA2Z4 (M = Ca, Sr, or Ba; A = In or Tl; Z = S, Se, or Te) compounds. Our study reveals the quantum spin Hall phase in SrTl2Te4 and BaTl2Te4 with a nontrivial topological bandgap of 97 and 28 meV, respectively, under a hybrid functional approach with the inclusion of spin–orbit coupling. Remarkably, the Z2 topological invariant and the presence of gapless edge states further confirmed their nontrivial topological phase. In addition, we demonstrate the quantized spin Hall conductivity in SrTl2Te4, which stems from the non-zero Berry curvature. The topological phase transition is driven by SOC due to the band inversion between the Te-px+py and Tl-s orbitals around Γ. Interestingly, the nontrivial topological properties are robust against strain and preserved under an applied electric field. Finally, our research identifies that the emergent MA2Z4 monolayers have interesting topological properties and have great potential for experimental realization of future topological applications.

Measuring the Boundary Gapless State and Criticality via Disorder Operator.

The disorder operator is often designed to reveal the conformal field theory (CFT) information in quantum many-body systems. By using large-scale quantum MonteCarlo simulation, we study the scaling behavior of disorder operators on the boundary in the two-dimensional Heisenberg model on the square-octagon lattice with gapless topological edge state. In the Affleck-Kennedy-Lieb-Tasaki phase, the disorder operator is shown to hold the perimeter scaling with a logarithmic term associated with the Luttinger liquid parameter K. This effective Luttinger liquid parameter K reflects the low-energy physics and CFT for (1+1)D boundary. At bulk critical point, the effective K is suppressed but it keeps finite value, indicating the coupling between the gapless edge state and bulk fluctuation. The logarithmic term numerically captures this coupling picture, which reveals the (1+1)D SU(2)_{1} CFT and (2+1)D O(3) CFT at boundary criticality. Our Letter paves a new way to study the exotic boundary state and boundary criticality.

Topological Phonons and Thermoelectric Conversion in Crystalline Materials

AbstractTopological phononics, a fascinating frontier in condensed matter physics, holds great promise for advancing energy‐related applications. Topologically nontrivial phonons typically possess gapless edge or surface states. These exotic states of lattice vibrations, characterized by their nontrivial topology, offer unique opportunities for manipulating and harnessing energy transport. The exploration of topological phonons opens new avenues in understanding and controlling thermal transport properties, with potential applications in fields such as thermoelectric materials, phononic devices, and waste heat recovery. Here, an overview of concepts such as Berry curvature and topological invariants, along with the applications of phonon tight‐binding method and nonequilibrium Green's function method in the field of topological phononics is provided. This review encompasses the latest research progress of various topological phonon states within crystalline materials, including topological optical phonons, topological acoustical phonons, and higher‐order topological phonons. Furthermore, the study delves into the prospective applications of topological phonons in the realm of thermoelectric conversion, focusing on aspects like size effects and symmetry engineering.

Imaging the Breakdown and Restoration of Topological Protection in Magnetic Topological Insulator MnBi2Te4.

Quantum anomalous Hall (QAH) insulators transport charge without resistance along topologically protected chiral 1D edge states. Yet, in magnetic topological insulators to date, topological protection is far from robust, with zero-magnetic field QAH effect only realized at temperatures an order of magnitude below the Néel temperature TN, though small magnetic fields can stabilize QAH effect. Understanding why topological protection breaks down is therefore essential to realizing QAH effect at higher temperatures. Here a scanning tunneling microscope is used to directly map the size of exchange gap (Eg,ex) and its spatial fluctuation in the QAH insulator 5-layer MnBi2Te4. Long-range fluctuations of Eg,ex are observed, with values ranging between 0 (gapless) and 70 meV, appearing to be uncorrelated to individual surface point defects. The breakdown of topological protection is directly imaged, showing that the gapless edge state, the hallmark signature of a QAH insulator, hybridizes with extended gapless regions in the bulk. Finally, it is unambiguously demonstrated that the gapless regions originate from magnetic disorder, by demonstrating that a small magnetic field restores Eg,ex in these regions, explaining the recovery of topological protection in magnetic fields. The results indicate that overcoming magnetic disorder is the key to exploiting the unique properties of QAH insulators.

Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles

Topology plays a crucial role in many physical systems, leading to interesting states at the surface. A paradigmatic example is the Chern number defined in the Brillouin zone that leads to robust gapless edge states. Here we introduce the reduced Chern number, defined in the subregions of Brillouin zone, and construct a family of Chern dartboard insulators with quantized reduced Chern numbers but with trivial bulk topology. Chern dartboard insulators are protected by the mirror symmetries and exhibit distinct pseudospin textures, including (anti)skyrmions, inside the sub-Brillouin zone. These Chern dartboard insulators host exotic gapless edge states, such as Möbius fermions and midgap corner states, and can be realized in the photonic crystals. Our work opens up new possibilities for exploring sub-Brillouin zone topology and nontrivial surface responses in topological systems.

Topological phase transitions and flat bands on an islamic lattice

We construct an islamic lattice by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-NN hoppings. This model supports abundant quantum phases for various values of filling fractions. At filling, Chern insulator (CI) phases with Chern numbers and a zero-Chern-number topological insulator (ZCNTI) phase exist. At filling, several CI phases with Chern numbers and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbers and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at and fillings. Phase diagrams for filling fractions , , and are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number emerge in this model and exactly flat band is found at 3/8 filling. We further investigate fractional Chern insulate state when hard-core bosons fill into this TFB model.

Gapless edge states localized to odd/even layers of AA′-stacked honeycomb multilayers with staggered AB-sublattice potentials

In honeycomb multilayers with staggered AB-sublattice potentials, we predict gapless edge states localized to either of the odd and the even layers for the AA^{prime } stacking order in which the sublattice-pseudospin polarizations of adjacent layers are antiparallel. Gaps in the projected layer-pseudospin spectrum suppress interlayer hopping between odd and even layers. The layer-valley Chern number corresponding to the edge states was obtained by decomposing the occupied state into two layer-pseudospin sectors by using a projected layer-pseudospin operator. For the AB^{prime } stacking, the sublattice-pseudospin polarizations of adjacent layers are antiparallel, but the layer-pseudospin spectrum gap closes at the interface of the topologically different states, leading to gapped edge states. For the AA and AB stackings where the sublattice-pseudospin polarizations of the adjacent layers are parallel, the gapless edge states corresponding to quantum valley Hall states are evenly distributed across the layers.

Utilizing Topological Insulator Two-Dimensional Bismuth for Ultrasensitive Acoustic Detection.

Topological insulators (TIs) are characterized by a full insulating gap in the bulk and gapless edge or surface states, which have attracted tremendous attention. 2D Bi (110), as a typical TI, is of particular interest due to its low symmetry structure and topologically protected and spin-momentum-locked Dirac surface states. However, the material's potential applications are hindered by difficulties in fabrication, due to its strong semi-metallic bonding and poor stability. In this study, a novel electrochemical intercalation method for the fabrication of ultrathin Bi (110) nanosheets with the highest yield ever reported is presented. These nanosheets are stabilized through cathodic exfoliation in a reductive environment and further modification with polymer ionic liquids. The versatility of these nanosheets is demonstrated by fabricating flexible acoustic sensors with ultrahigh sensitivity. These sensors can even detect sounds as quiet as 45dB. Furthermore, these sensors are utilized for acoustic-to-electric energy conversion and information transfer. This work offers a promising approach for scalable fabrication and preservation of ultrathin 2D TI Bi (110) nanosheets and paves the way for their integration into smart devices.

Rational design of quantum spin Hall phase in type-III van der Waals heterostructures

Van der Waals heterostructures (vdWHs) are effective platforms for exploring various attractive topological phases. Here, based on the low-energy effective k·p model, we propose that the type-III vdWHs, which were previously considered as only belonging to trivial metallic phases, can realize the nontrivial quantum spin Hall (QSH) effect. We reveal that the band inversion of such a QSH phase is attributed to the band alignment of momentum space matching, i.e., the conduction band minimum and valence band maximum located at the same point in momentum space near the Fermi level. Moreover, using first-principles calculations, we show that the Mg(OH)2/Ga2O2 heterobilayer, a typical type-III vdWH with high thermodynamic stability, is an ideal candidate for achieving our strategy. We further calculate the helical gapless edge states and quantized spin Hall conductance, which are visible inside the global bandgap, thus facilitating the experimental observation. Our work offers a promising pathway for realizing the QSH phase in natural materials.

Exploring the electronic and topological properties of single-layer VP2X2Y2 (XY= AlS, BTe, GaS): Valley polarization and quantum anomalous Hall state

The concept of manipulating the valley degree of freedom has garnered growing attention in both fundamental scientific research and emerging applications. In this study, first-principles calculations are performed on the electronic properties and structural stability of single-layer (SL) VP2X2Y2 (XY = AlS, GaS and BTe). All three materials have been identified as ferromagnetic semiconductors, with their conduction band minimum situated at the K point. VP2Al2S2 and VP2Ga2S2, in particular, possess easy out-of-plane magnetization, thus facilitating a pronounced valley polarization observed in the bottom conduction band. Applying compressive strain can further modulate the orbital character of the bands as well as the valley polarization, leading to a transition from the bottom conduction band to the top valence band. Intriguingly, the strained SL VP2Al2S2 and VP2Ga2S2 exhibit a quantum anomalous Hall state with a nonzero Chern number of C=1, which is further evidenced by a quantized Hall conductance and a chiral gapless edge state connecting the valence and conduction bands. Thus, the unique electronic and topological properties of SL VP2X2Y2 suggest potential applications in the fields of electronics, spintronics, and valleytronics.

Observation of nontrivial Zak phase induced topological states in glow discharge plasma

Plasma blackout, which contains ablative impurities, strongly attenuates the signal of the reentry spacecraft. Traditional methods focus on mitigating electron densities and impurities around the antenna, and metamaterial-based electromagnetic methods have yet to be proven experimentally. We simulate the plasma blackout problem using laboratory plasma supported by gas discharge technology. Alumina pillars are embedded in the plasma background to form plasma photonic crystals, while topological phase transitions are achieved by shrinking and expanding pillars within a unit cell. The topological edge states (TESs) that are insensitive to weak impurities in the transport path are verified theoretically and experimentally. We introduce the glide-reflection (GR) symmetry in the nontrivial lattices to obtain the gapless edge states, which are exclusively observed in the acoustic systems. Meanwhile, the Δω of the gapless TES increases with the electron densities, ensuring a wide communication bandwidth. Furthermore, the strong coupling of heterostructure with GR symmetry in plasma photonic crystals is elucidated. Our work not only provides a new approach to the blackout communication problem but can also serve as a nascent experimental platform to investigate topological electromagnetic phenomena.

Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.

Femtosecond laser upgrading the quality of bismuth films to enhance ultra-broadband photodetection.

Topological insulator bismuth has attracted considerable attention for the fabrication of room-temperature, wide bandwidth, and high-performance photodetectors due to the gapless edge state and insulating bulk state properties. However, both the photoelectric conversion and carrier transportation of the bismuth films are extremely affected by the surface morphology and grain boundaries to limit optoelectronic properties further. Here, we demonstrate a strategy of femtosecond laser treatment for upgrading the quality of bismuth films. After the treatment with proper laser parameters, the measurement of average surface roughness can be reduced from Ra = 44 nm to 6.9 nm, especially with accompany of the evident grain boundary elimination. Consequently, the photoresponsivity of the bismuth films increases approximately 2 times within an ultra-broad spectrum range from the visible to mid-infrared. This investigation suggests that the femtosecond laser treatment can help to benefit the performance of topological insulator ultra-broadband photodetectors.

Non-trivial topological crossover in functionalized AlBi monolayer

The incorporation of spin–orbit coupling and altering the orbital composition close to the Fermi level via functionalization are two essential techniques for exploring topological insulators. We have shown through an ab-initio study that spin–orbit coupling transforms the fluorinated AlBi honeycomb monolayer from a normal insulator to a topological insulator with a large band gap. The combined effects of functionalization and spin–orbit coupling led to a band-inversion in the AlBi honeycomb monolayer. This monolayer has a nontrivial topological character, as confirmed by the reported s/p band inversion in the band structures, the Z2 topological invariant and the clear demonstration of the topological gapless edge states bridging the valence band and conduction band. In addition, it is found that compressing the fluorinated AlBi lattice by about 3 % triggered the topological phase transition, which results in a normal state. The reported nontrivial band gap of 0.36 eV in the fluorinated AlBi monolayer is sufficient to make them a suitable consideration for spintronic devices at room temperature.